Stochastic Spline-Collocation Method foe Constrained Optimal Control Problem Governed by Random Elliptic PDE
Stochastic Spline-Collocation Method foe Constrained Optimal Control Problem Governed by Random Elliptic PDE
复制标题
随机椭圆偏微分方程约束最优控制问题的随机样条配置方法
DOI：
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发表时间：
2017
影响因子：
1.1
通讯作者：
W.B. LIU
中科院分区：
文献类型：
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作者：
BENXUE GONG;LIANG GE;TONGJUN SUN;WANFANG SHEN;W.B. LIU
本文研究了具有随机场系数的椭圆型偏微分方程解的最优控制问题的随机样条配置逼近格式。我们得到了最优控制问题最优性的充要条件，并通过对空间有限元方法和随机样条配点法对概率空间的离散化，建立了逼近最优系统的方案。我们进一步研究了Smolyak近似格式，它们对于依赖于适度大的随机变量的光滑问题是有效的配置策略。对于更一般的状态可能是非光滑的控制问题，对于某些区域的随机变量，我们采用区域分解策略将随机空间划分为光滑和非光滑部分，然后分别应用Smolyak格式和Spline逼近。给出了状态、协同状态和控制变量的先验误差估计。文中给出了数值算例来说明我们的理论结果。
In this paper, we investigate a stochastic spline-collocation approximation scheme.for an optimal control problem governed by an elliptic PDE with random field coefficients. We.obtain the necessary and sufficient optimality conditions for the optimal control problem and.establish a scheme to approximate the optimality system through the discretization with respect.to the spatial space by finite elements method and the probability space by stochastic splinecollocation method. We further investigate Smolyak approximation schemes, which are effective.collocation strategies for smooth problems that depend on a moderately large number of random.variables. For more general control problems where the state may be non-smooth with respect.to the random variables in some areas, we adopt a domain decomposition strategy to partition.the random space into smooth and non-smooth parts and then apply Smolyak scheme and spline.approximation respectively. A priori error estimates are derived for the state, the co-state and.the control variables. Numerical examples are presented to illustrate our theoretical results.